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References Space Groups for Solid State Scientists , G. Burns and A . M. Glazer

Physics 590 “International Tables of Crystallography”. “Everything you wanted to know about beautiful flies , but were afraid to ask.”. Schönflies. Gordie Miller (321 Spedding ). Proposed Plan What basic information is found on the space group pages…

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References Space Groups for Solid State Scientists , G. Burns and A . M. Glazer

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  1. Physics 590 “International Tables of Crystallography” “Everything you wanted to know about beautiful flies, but were afraid to ask.” Schönflies Gordie Miller (321 Spedding) Proposed Plan What basic information is found on the space grouppages… Stoichiometry of the unit cell (Wyckoff sites) Site (point) symmetry of atoms in solids Solid-solid phase transitions (group-subgroup relationships) Diffraction conditions – what to expect in a XRD powder pattern. References Space Groups for Solid State Scientists, G. Burns and A. M. Glazer (Little mathematical formalism; prose style) International Tables for Crystallography (http://it.iucr.org/) Bilbao Crystallographic Server (www.cryst.ehu.es) (Comprehensive resources for all space groups)

  2. What can we learn from the International Tables? BaFe2As2 Space Group: I4/mmm Lattice Constants:a = 3.9630 Å c = 13.0462 Å Asymmetric Unit: Ba (2a): 0 0 0 Fe (4d): ½ 0 ¼ As (4e): 0 0 0.3544 c Intensity (Arb. Units) (hkl) Indices h + k + l= even integer (2n) (013) (116) (200) (112) (213) b a (015) (004) (028) (215) (011) (002) 2θ (Cu Kα)

  3. Typical Space Group Pages… Symbolism Diffraction Extinction Conditions Point Symmetry Features Stoichiometry Structure of Unit Cell Subgroup/Supergroup Relationships

  4. Symbolism Point Group of the Space Group Crystal System Space Group Molecules Solids S2= h C2 NOTE: , (z) y In Schönflies notation, what does the symbol S2 mean? (x,y,z) C2 rotation followed by shC2 axis S2= inversion x In International notation, what does the symbol mean? (z) y 2-fold (C2) rotation followed by inversion ( ) (x,y,z) , Why are the symbols S2 and not used? x

  5. Symbolism: Crystal Systems What rotational symmetries are consistent with a lattice (translational symmetry)? C1 C2 (2π/2) C3 (2π/3) C4 (2π/4) C6 (2π/6) •  = angle between b and c •  = angle between a and c •  = angle between a and b c a b

  6. Symbolism: Bravais Lattices 7 Crystal Systems = 7 Primitive Lattices (Unit Cells): P (rhombohedral) “Centered Lattices” •  = angle between b and c •  = angle between a and c •  = angle between a and b ? c a b I F C B A Body- (All) Face- Base-

  7. Symbolism: Point Groups • Schönflies • Notation

  8. Symbolism: Crystallographic Point Groups 32 Point Groups Allowed Rotations = C1C2C3C4C6 Yes: Laue Groups b c • m2m or mm2 a b c c a b a+b a–b

  9. Questions for Friday… For the following space group symbol What is the crystal class? What is the lattice type? Which face(s) are centered? (c) What is the point group of the space group using Schönflies notation? (d) Does the point group contain the inversion operation? Cmm2 Orthorhombic Base (C)-centered ab-faces C2v No

  10. Symbolism: Crystallographic Point Groups (cont.) c a b • 31m • 312 c a b a b c

  11. Symbolism: Symmetry Operations (4h = 4/mmm = 4/m 2/m 2/m) Improper Rotations Proper Rotations – + 4 / m mm , , – + , , – – + + 33 Matrices: , , – – + + – + y , , x – + (Determinant = +1) (Determinant = -1)

  12. Symmorphic Space Groups General Position Special Positions Point Group= {Symmetry operations intersecting in one point} (32) Space Group = {Essential Symmetry Operations}  {Bravais Lattice} (230)

  13. Symmorphic Space Groups Ba (2a): 4/mmm (D4h) Fe (4d): m2(D2d) As (4e): 4mm (C4v) General Position Special Positions “Ba2Fe4As4” Z = 2 Space Group: I4/mmm Lattice Constants:a = 3.9630 Å c= 13.0462 Å Asymmetric Unit: Ba (2a): 0 0 0 Fe (4d): ½ 0 ¼ As (4e): 0 0 0.3544 BaFe2As2

  14. Space Groups (230) SymmorphicSpace Groups (73): {Essential Symmetry Operations} is a group. Point Group of the Space Group Nonsymmorphic Space Groups (157): {Essential Symmetry Operations} is a not a group.

  15. Space Group Operations: Screw Rotations and Glide Reflections Screw Rotations: Rotation by 2/n (Cn) then Displacement j/n lattice vector || Cn axis (allowed integers j = 1,…, n–1) Symbol = nj 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, 65 I41/amd 4/mmm Point Group of the Space Group Glide Reflections: Reflection then Displacement 1/2 lattice vector || reflection plane Axial:a, b, c (lattice vectors = a, b, c) Diagonal:n (vectors = a+b, a+c, b+c) Diamond: d (vectors = (a+b+c)/2, (a+b)/2, (b+c)/2, (b+c)/2) P42/ncm Nonsymmorphic Space Groups (157)

  16. The Origin! Si: 0, 0, 0

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